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Theorem r19.3rm 3354
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
Hypothesis
Ref Expression
r19.3rm.1 𝑥𝜑
Assertion
Ref Expression
r19.3rm (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem r19.3rm
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2147 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1840 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2147 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1840 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 biimt 239 . . . 4 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
6 df-ral 2360 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 r19.3rm.1 . . . . . 6 𝑥𝜑
8719.23 1611 . . . . 5 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
96, 8bitri 182 . . . 4 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
105, 9syl6bbr 196 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 119 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
122, 11sylbir 133 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285  wnf 1392  wex 1424  wcel 1436  wral 2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-cleq 2078  df-clel 2081  df-ral 2360
This theorem is referenced by:  r19.28m  3355  r19.3rmv  3356  r19.27m  3361  indstr  8990
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